Solution - Nonlinear equations

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     100*x^20-(50)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (22•52x20) -  50  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   100x²⁰ - 50  =   50 • (2x²⁰ - 1) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  2x²⁰ - 1 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check :  2  is not a square !!

Ruling : Binomial can not be factored as the
difference of two perfect squares

Equation at the end of step  3  :

  50 • (2x20 - 1)  = 0 

Step  4  :

Equations which are never true :

 4.1      Solve :    50   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Solving a Single Variable Equation :

 4.2      Solve  :    2x²⁰-1 = 0 

 Add  1  to both sides of the equation : 
                      2x²⁰ = 1
Divide both sides of the equation by 2:
                     x²⁰ = 1/2 = 0.500
                     x  =  20th root of (1/2) 

 The equation has two real solutions  
 These solutions are  x = 20th root of ( 0.500) = ± 0.96594  
 

Two solutions were found :